Equivalence Relations.

Report
Equivalence Relations:
Selected Exercises
Equivalence Relation
• Let E be a relation on set A.
• E is an equivalence relation if & only if it is:
– Reflexive
– Symmetric
– Transitive.
• Examples
– a E b when a ≡ b ( mod 5 ). (Over N)
– a E b when a is a sibling of b. (Over humans)
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Equivalence Class
• Let E be an equivalence relation on A.
• We denote aEb as a ~ b. (sometimes, it is denoted a ≡ b )
• The equivalence class of a is { b | a ~ b }, denoted [a].
• What are the equivalence classes of the example equivalence
relations?
• For these examples:
– Do distinct equivalence classes have a non-empty intersection?
– Does the union of all equivalence classes equal the underlying set?
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Partition
A partition of set S is a set of nonempty subsets,
S1, S2, . . ., Sn, of S such that:
1. i j ( i ≠ j  Si ∩ Sj = Ø ).
2. S = S1 U S2 U . . . U Sn.
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Equivalence Relations & Partitions
Let E be an equivalence relation on S.
• Thm. E’s equivalence classes partition S.
• Thm. For any partition P of S, there is an equivalence
relation on S whose equivalence classes form partition P.
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E’s equivalence classes partition S.
1.
[a] ≠ [b]  [a] ∩ [b] = Ø.
Proof by contradiction:
Assume [a] ≠ [b]  [a] ∩ [b] ≠ Ø:
(Draw a Venn diagram)
Without loss of generality, let c  [a] - [b]. Let d  [a] ∩ [b].
We show that c  [b] (which contradicts our assumption above)
2.
1. c ~ d
( c, d  [a] )
2. d ~ b
( d  [b] )
3. c ~ b
( c ~ d  d ~ b  E is transitive )
The union of the equivalence classes is S.
Students: Show this use pair proving in class.
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For any partition P of S, there is an equivalence relation
whose equivalence classes form the partition P.
Prove in class.
1. Let P be an arbitrary partition of S.
2. We define an equivalence relation whose
equivalence classes form partition P.
(Students: Show this (use pair proving) in class)
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Exercise 40
a)
What is the equivalence class of (1, 2) with respect
to the equivalence relation given in Exercise 16?
Exercise. 16:
Ordered pairs of positive integers such that
( a, b ) ~ ( c, d )  ad = bc.
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Exercise 40 a) Answer
( a, b ) ~ ( c, d )  ad = bc  a/b = c/d
[ ( 1, 2 ) ] = { ( c, d ) | ( 1, 2 ) ~ ( c, d ) }
= { ( c, d ) | 1d = 2c  c/d = ½ }.
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Exercise 40 continued
b) Interpret the equivalence classes of the equivalence
relation R in Exercise 16.
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Exercise 40 continued
b) Interpret the equivalence classes of the equivalence
relation R in Exercise 16.
Answer
Each equivalence class contains all (p, q), which, as
fractions, have the same value (i.e., the same
element of Q+).
(The fact that 3/7 = 15/35 confuses some small children.)
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Exercise 50
• A partition P’ is a refinement of partition P when
x  P’ y  P x  y. (Illustrate.)
• Let partition P consist of sets of
people living in the same US state.
• Let partition P’ consist of sets of
people living in the same county of a state.
• Show that P’ is a refinement of P.
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Exercise 50 continued
It suffices to note that:
Every county is contained within its state:
No county spans 2 states.
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Exercise 62
Determine the number of equivalent relations on a set
with 4 elements by listing them.
How would you represent the equivalence relations
that you list?
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End 8.5
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10
Suppose A   & R is an equivalence relation on A.
Show f X f: A  X such that a ~ b  f( a ) = f( b ).
Proof.
1. Let f : A  X, where
1. X = { [a] | [a] is an equivalence class of R }
2.
a f (a ) = [a].
2. Then, a b a ~ b  f( a ) = [a] = [b] = f( b ).
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Exercise 20
• Let P be the set of people who visited web page W.
• Let R be a relation on P: xRy  x & y visit the same
sequence of web pages since visiting W until they exit the
browser.
•
Is R an equivalence relation?
• Let s( p ) be the sequence of web pages p visits since
visiting W until p exits the browser.
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Exercise 20 continued
• That is, xRy means s( x ) = s( y ).
• x xRx: R is reflexive.
Since x s( x ) = s( x ).
• x y ( xRy  yRx ): R is symmetric.
Since s( x ) = s( y )  s (y ) = s( x ).
• x y z ( ( xRy  yRz )  xRz ): R is transitive.
Since ( s( x ) = s( y )  s( y ) = s( z ) )  s( x ) = s( z ).
• Therefore, R is an equivalence relation.
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Exercise 30
What are the equivalence classes of the bit strings for the
equivalence relation of Exercise 11?
Ex. 11: Let S = { x | x is a bit string of ≥ 3 bits. }
Define xRy such that x agrees with y on the left 3 bits
(e.g., 10111 ~ 101000).
a) 010
b) 1011
c) 11111
d) 01010101
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Exercise 30 Answer
•
010
(answer: all strings that begin with 010)
•
1011
(answer: all strings that begin with 101)
•
11111
(answer: all strings that begin with 111)
•
01010101
(answer: all strings that begin with 010)
How many equivalence classes are there?
Copyright © Peter Cappello
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